Simple, Strict, Proper, and Directed: Comparing Reachability in Directed and Undirected Temporal Graphs

2025-01-20

AI Summary

This paper systematically investigates the expressive power hierarchy of directed and undirected time-varying graphs under reachability semantics. Addressing three fundamental computational properties—support, reachability, and indirect reachability—the work introduces *directionality* into temporal graph classification for the first time, constructing six directed and six undirected models spanning strict/non-strict, canonical/arbitrary, and simple/multi-labeled variants. Through equivalence and convertibility analysis, it establishes that directed models form a single-chain partial order, whereas undirected models exhibit a two-chain incomparable structure; moreover, every undirected model is unidirectionally simulatable by some directed model, but not vice versa. Leveraging three formal equivalence notions—support equivalence, reachability equivalence, and induced-reachability equivalence—combined with constructive reductions and counterexample modeling, the paper fully characterizes the expressive power preorder among all twelve model classes and proposes a cross-directional unidirectional simulation theorem, providing a formal foundation for complexity transfer across temporal graph algorithms.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Graphical Models
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Web Mining and Content Analysis: Models for Web evolution

Abstract

We present the first comprehensive analysis of temporal settings for directed temporal graphs, fully resolving their hierarchy with respect to support, reachability, and induced-reachability equivalence. These notions, introduced by Casteigts, Corsini, and Sarkar, capture different levels of equivalence between temporal graph classes. Their analysis focused on undirected graphs under three dimensions: strict vs. non-strict (whether times along paths strictly increase), proper vs. arbitrary (whether adjacent edges can appear simultaneously), and simple vs. multi-labeled (whether an edge can appear multiple times). In this work, we extend their framework by adding the fundamental distinction of directed vs. undirected. Our results reveal a single-strand hierarchy for directed graphs, with strict&simple being the most expressive class and proper&simple the least expressive. In contrast, undirected graphs form a two-strand hierarchy, with strict&multi-labeled being the most expressive and proper&simple the least expressive. The two strands are formed by the non-strict&simple and the strict&simple class, which we show to be incomparable. In addition to examining the internal hierarchies of directed and of undirected graph classes, we compare the two. We show that each undirected class can be transformed into its directed counterpart under reachability equivalence, while no directed class can be transformed into any undirected one. Our findings have significant implications for the study of computational problems on temporal graphs. Positive results in more expressive graph classes extend to weaker classes as long as the problem is independent under reachability equivalence. Conversely, hardness results for a less expressive class propagate to stronger classes. We hope these findings will inspire a unified approach for analyzing temporal graphs under the different settings.

Problem

• Directed Temporal Graphs
• Undirected Temporal Graphs
• Computational Problems

Innovation

• Time-Varying Graphs
• Directed vs Undirected Graphs
• Accessibility Hierarchy